Sorry for being pedantic, but I want to clarify should anyone else happen upon this. Intuitionistic logic does not affirm double negation elimination (which is equivalent to the Law of Excluded Middle). I assume this is what you meant, but there is nothing preventing discussing “double negation” in intuitionistic logic.

Also, in classical logic, direct proof of negation (i.e. suppose P, get a contradiction, conclude not P) and proof by contradiction (i.e. suppose not P, get a contradiction, therefore not not P, use double negation elimination to conclude P) are both typically referred to as proof by contradiction. The latter is clearly not intuitionistically valid because the final step relies upon applying excluded middle.

]]>I’m happy you posted. This work is a bit old, and I have a better treatment on the way, but in the mean time:

* I agree with what you say about intuitionistic logic, and I’m trying to do a similar thing with mathematics in general (for example, point-sets are not a good mental model for space; see the Wikipedia entry on point-free topology).

* Modal logic adds possibility and necessity; I was not aware that it added tense. It is thus an approximation of fuzzy logic, which uses non-binary truth values.

* You state that “It seems ‘might be’ is not the same as either {true, false} or {}, though the Catuskoti mentioned here, like Modal Logic, seeks to help logic live up better to the world and human language by adding more options for truth values.” Modal logic is probably better seen as adding modifiers to truth values rather than adding entirely new truth values. The catuskoti definitely adds two new truth values; both and neither.

* With respect to the catuskoti, it is a very old logic, and I am not an expert in it. However, I have studied the philosophical context from which it arises. That context was strongly influenced by subjective observations (how do we know things) and belief that things in the world are not singular.

So I think {} could be treated as the truth value Unknown; this is done in some ternary logics such as Priest or Kleene logics. It may be a good fit for intuitionistic logic, although I do not know the formal definition of vacuity in that logic. This accommodates subjectivity within Boolean logic.

To accommodate non-singular predicates, as the madhyamika logicians who invented the catuskoti would want to do, is what requires the truth value {true,false}. As far as I can tell, though, I am alone in backing that hypothesis (the professional logicians I’ve written to about it have not responded).

Personally, I think logic as taught in our schools should have an Unknown value. In that context, an affirming negation operates as usual, turning True and False values into each other, but there is also an opportunity for non-affirming negation, which turns True and False values into the value Unknown. Whether we need the additional complexity of fuzziness/modality and compound predicates probably depends on the abilities of those to whom it is taught.

]]>I learned about this mostly through inquiry in basic-level classes, so I’m unfamiliar with the relation between intuitionistic logic, modal logic, and catuskoti.

However, from my experience, which perhaps someone can clarify or add on, it seems that modal logic seeks to generally help logic fit better into natural language by adding tense (will be, was) as well as possibility (might be).

It seems “might be” is not the same as either {true, false} or {}, though the Catuskoti mentioned here, like Modal Logic, seeks to help logic live up better to the world and human language by adding more options for truth values.

In terms of the specific situations addressed, such as ambiguous (vacuous) situations, it seems Catuskoti and Intuitionistic Logic both seek to avoid claiming vacuous truth, though Intu.. simply avoids stating it (and situations that would imply it, such as double negations), while Catuskoti positively states neither True nor False when nothing can be claimed, correct? ]]>

Also, as a Buddhist, job well done!

]]>There is an assumption in this argument that would not be accepted by intuitionists: “x may lie on the boundary between two intervals”.

For the intuitionist argument to be plausible, the interval between 0 and 1 cannot consist of an infinite number of points. Mereotopology seems to be the standard answer, which takes ‘part’ as primitive instead of ‘point’. Intuitively, a space of a given dimensionality cannot consist of parts of a lower dimensionality, as is the case with infinitely many points creating a line.

The real numbers can represent points of division of a continuous interval, but since they have zero width, they are nominal (merely imputed) on the interval. I continue to think that all decimal representations which are different from one another will divide that interval at different places.

This point of view might be able to be summed up topologically by saying that all intervals are open intervals, but in this case open intervals are said to be touching if they share a boundary (which in the linear case is a point). That feels desirable for intuitionists, because it avoids the topological paradoxes of objects with open/closed boundaries being unable to be in contact with one another. I am not sure if doing this creates other difficulties, but there are no associated paradoxes of which I am aware.

]]>Let’s begin the challenge by writing the two numbers in decimal form:

1) $latex x_1 = 0.9\overline{9} = \sum\limits_{n=1}^{\infty} \frac{9}{10^n} &s=1$

2) $latex x_2 = 1.0 = 1+\sum\limits_{n=1}^{\infty} \frac{0}{10^n} &s=1$

3) $latex x_2 – x_1 = 1.0 – 0.9\overline{9} = 1-\sum\limits_{n=1}^{\infty} \frac{9}{10^n} &s=1$

If we write out the continuing approximation of this series for n=1,2,3,…, we have:

3) $latex 1, 0.1, 0.01, 0.001, 0.0001, … &s=1$

To express this as a series, we have:

4) $latex 1 + \sum\limits_{n=1}^{\infty} \frac{1}{10^n} – \frac{1}{10^(n-1)} &s=1$

I think this is a good and explicit representation: it is explicit in the sense that it does not express a completed infinity. I feel that the overbar notation, understood as an infinite decimal string, leads to mathematical errors. However, limit series are inconvenient to write, so the following shorthand for exactly the equations given above seems reasonable:

4) $latex x_2 – x_1 = 1.00 – 0.9\overline{9} = 1.00 – 0.\overline{9}9 = 0.\overline{0}1 &s=1$

I have not seen the overbar used like this before, and in fact, I am not sure if the overbar notation has a formal definition (it was explained to me as “repeat whatever decimals are under the bar *infinitely*“). However, it suggests that a decent definition (for an intuitionist) would be something like:

5) $latex \overline{x} := \sum\limits_{n=1}^{\infty} {x^n} = x, xx, xxx, … &s=1$

Of course, this needs to be understood as a string manipulation, since it will have different effects whether it is used before or after the decimal point. Its use here under a summation is important because it indicates the importance of the loop variable, $latex n &s=1$. To reiterate the issue that began this thread, the steps toward convergence of the series are essential for intuitionist philosophers, because we never reach a completed infinity. For that reason, please reflect on the necessity of the following inequality for mathematicians who do not believe in a completed infinity:

6) $latex 0.9\overline{9} \neq 0.\overline{9} &s=1$

Although in some sense they can be said to be equal *at infinity*, they are not equal for any finite-length expansion of the index *n* (which is implicit in the overbar notation).

The second response is a bit more challenging for me. Because I am defending the uniqness of $latex 0.9\overline{9} &s=1$ and $latex 1.0 &s=1$, I was asked for the decimal representation of $latex 1 – 0.9\overline{9} &s=1$. If I cannot give a decimal representation of that number, then I must concede that not all numbers have decimal representations (which I probably have to concede for irrational numbers anyway, since that seems to require decimal strings which have a length corresponding to a completed infinity). That said, it is a good question: is there a decimal representation of $latex 1 – 0.9\overline{9} &s=1$, and if so, what is it?

]]>I am one of those people whose interests lie “at the crossroads of psychology, linguistics, logic, mathematics, and philosophy”. This is why I started to read the electronic version of your book.

Like you, I see the need to connect “the real” with “the mental”, but I prefer to do this by synthesizing these two contradictory views. I think that most of my human fellows tend to ignore the cognitive issue of scale and limit their analysis to the simple notion of part/whole relation in the single-scale world. Acknowledging existence of scales helps to see that mental phenomena are just an observation of smaller-scale phenomena by a larger-scale collection. Consider the issue of light duality as an example – the wave behavior of light is an artifact of the large-scale observation of corpuscular behavior at the smaller scale. These are “views” of light from two different scales (not perspectives but scales). See for example Victor Stenger’s video titled “Victor Stenger – The Future of Naturalism Interview”:

https://www.youtube.com/watch?v=Ayst78EjjS0

Just like you, I am working on the theory I developed in my PhD thesis in which I presented a naturalistic interpretation of cognition. I was awarded my degree earlier this year. I wonder if it would be possible to know a bit more about you, the author of the book. I am looking both for someone with whom I could engage in a meaningful discussion and for a mentor.

My regards, Wes

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